Portfolio optimisation with small transaction costs

1 / 26 Portfolio optimisation with small transaction costs an engineer s approach Jan Kallsen based on joint work with Johannes Muhle-Karbe and Paul Krühner New York, June 5, 2012 dedicated to Ioannis Karatzas

2 / 26 Outline 1 Introduction 2 Asymptotically optimal portfolios for small costs 3 Multivariate extension

3 / 26 Outline 1 Introduction 2 Asymptotically optimal portfolios for small costs 3 Multivariate extension

4 / 26 Portfolio choice under transaction costs using duality Cvitanić & Karatzas (1996): Loewenstein (2000):

Shadow price processes in optimisation The basic principle Goal: max E(u(V T (ϕ))) ϕ V (ϕ) wealth under transaction costs Ex. S in [S, S] such that optimiser ϕ maximises ϕ E(u(v 0 + ϕ ST )), V T (ϕ ) = v 0 + ϕ ST. bid 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 0.0 0.2 0.4 0.6 0.8 1.0 t 5 / 26

6 / 26 What about explicit results using shadow prices? Cvitanić & Karatzas (1996): but cf. K. & Muhle-Karbe (2009), Kühn & Stroh (2010), Gerhold et al. (2011), Herczegh & Prokaj (2011), Choi et al. (2012)

7 / 26 What about explicit results using shadow prices? Problem: explicit results are very rare. (e.g. Davis & Norman 1990,... ) Way out: look at asymptotics for small transaction costs. (Constantinides 1986, Whalley & Wilmott 1997, Atkinson & Al-Ali 1997, Janeček & Shreve 2004, Rogers 2004,... )

8 / 26 Outline 1 Introduction 2 Asymptotically optimal portfolios for small costs 3 Multivariate extension

9 / 26 Our setup The portfolio optimisation problem Want to maximise ϕ E( exp( pv T (ϕ))) V (ϕ) wealth process under transaction costs bid price S = (1 ε)s, ask price S = (1 + ε)s Why exponential utility u(x) = exp( px)? It allows for random endowment (= hedging problem) by a measure change. Solution does not depend on initial wealth. Utility on R better suited for hedging than utilities on R+. Exponential utility often leads to simple structure.

10 / 26 Our setup The traded asset Traded asset ds t = b t dt + σ t dw t univariate continuous otherwise rather arbitrary Frictionless optimiser ϕ is assumed to be known.

11 / 26 Shadow price approach Look for optimal strategy ϕ ε, shadow price process S ε, dual martingale Z ε, which satisfy Z T = u (v 0 + ϕ ε Sε T ), Z ε martingale, Z ε S ε martingale, ϕ ε changes only when S t ε {S t, S t }.

12 / 26 Optimality of ϕ ε from an engineer s perspective For any competitor ψ we have E(u(V T (ψ))) E(u(v 0 + ψ Sε T )) ( ) E(u(v 0 + ϕ ε Sε T )) + E u (v 0 + ϕ ε Sε T )((ψ ϕ ε ) Sε T ) ( ) = E(u(V T (ϕ ε ))) + E Z T ((ψ ϕ ε ) Sε T ) = E(u(V T (ϕ ε ))). (Second inequality follows from u(y) u(x) + u (x)(y x).)

13 / 26 Approximate solution Look for approximately optimal strategy ϕ ε, approximate shadow price process S ε, approximate dual martingale Z ε, which satisfy Z T = u (v 0 + ϕ ε Sε T ) + O(ε), Z ε has drift o(ε 2/3 ), Z ε S ε has drift O(ε 2/3 ), ϕ ε changes only when S t ε {S t, S t }.

14 / 26 Approximate optimality of ϕ ε from an engineer s perspective For any competitor ψ ε with ψ ε ϕ = o(1) we have E(u(V T (ψ ε ))) E(u(v 0 + ψ ε Sε T )) ( ) E(u(v 0 + ϕ ε Sε T )) + E u (v 0 + ϕ ε Sε T )((ψ ε ϕ ε ) Sε T ) ( ) = E(u(V T (ϕ ε ))) + E Z T ((ψ ε ϕ ε ) Sε T ) + o(ε 2/3 ) = E(u(V T (ϕ ε ))) + o(ε 2/3 ). Compare with E(u(V T (ϕ ε ))) E(u(v 0 + ϕ ST )) = O(ε 2/3 ).

15 / 26 How to find an approximate solution The ansatz How to find ϕ ε, S ε, Z ε, no-trade bounds ϕ ±? Write ϕ ε = ϕ + ϕ, S ε = S + S, Z ε = Z (1 + K ), where ϕ, Z are optimal for the frictionless problem. Ansatz: S = f ( ϕ,... ), f (x) = αx 3 γx

How to find an approximate solution The solution This works for with S t = ( ) (γ t ϕ) 3 εst 3γ t x) 2 2p c γ t = 3 t S, 3εS t where d S, S t = ct S dt and d ϕ, ϕ t = c ϕ t dt. no-trade bounds ϕ ± t = ± 1 3εS = ± 3 t c ϕ t γ t 2p ct S c ϕ t certainty equivalent of utility loss ( ) T p E Z T 2 ( ϕ± t ) 2 d S, S T 0 16 / 26

How to find an approximate solution The solution This works for with S t = ( ) (γ t ϕ) 3 εst 3γ t x) 2 2p c γ t = 3 t S, 3εS t where d S, S t = ct S dt and d ϕ, ϕ t = c ϕ t dt. no-trade bounds ϕ ± t = ± 1 3εS = ± 3 t c ϕ t γ t 2p ct S c ϕ t certainty equivalent of utility loss ( ) T p E Z T 2 ( ϕ± t ) 2 d S, S T 0 17 / 26

How to find an approximate solution The solution This works for with S t = ( ) (γ t ϕ) 3 εst 3γ t x) 2 2p c γ t = 3 t S, 3εS t where d S, S t = ct S dt and d ϕ, ϕ t = c ϕ t dt. no-trade bounds ϕ ± t = ± 1 3εS = ± 3 t c ϕ t γ t 2p ct S c ϕ t certainty equivalent of utility loss ( ) T p E Z T 2 ( ϕ± t ) 2 d S, S T 0 18 / 26

19 / 26 Outline 1 Introduction 2 Asymptotically optimal portfolios for small costs 3 Multivariate extension

20 / 26 The multivariate case The setup Traded asset ds t = b t dt + σ t dw t with R d -valued S and multivariate Wiener process W (e.g. Akian et al. 1996, Liu 2004, Lynch & Tan 2006, Muthuraman & Kumar 2006, Atkinson & Ingpochai 2007, Goodman & Ostrov 2007, Law et al. 2009, Bichuch & Shreve 2012) Frictionless optimiser ϕ is assumed to be known.

21 / 26 The multivariate case The ansatz How to find ϕ ε, S ε, Z ε, no-trade region? Write ϕ ε = ϕ + ϕ, S ε = S + S, Z ε = Z (1 + K ), where ϕ, Z are optimal for frictionless problem. Ansatz: S i = f i ( ϕ,... ), i = 1,..., d ( ) εs f i (x) = (γi x) 3 3γi i x 2 No-trade region is parallelotop spanned by d vectors ±a 1..., ±a d R d.

The multivariate case The solution ( ) εs S i = (γi x) 3 3γi i x 2 works for γ = (γ 1,..., γ d ) R d d given by 2p 1 γ ij = 3 3εS i (c S c ϕ c S cij S ), ii where d S i, S j t = (c S ij ) tdt and d ϕ i, ϕ j t = (c ϕ ij ) tdt. no-trade parallelotop is spanned by a 1,..., a d R d, where a = (a 1,..., a d ) R d d is given by a ij = (γ 1 3εS ) ij = i 3 2p (cs c ϕ c S ) jj (c S ) 1 ij certainty equivalent of utility loss more complicated than in univariate case 22 / 26

The multivariate case The solution ( ) εs S i = (γi x) 3 3γi i x 2 works for γ = (γ 1,..., γ d ) R d d given by 2p 1 γ ij = 3 3εS i (c S c ϕ c S cij S ), ii where d S i, S j t = (c S ij ) tdt and d ϕ i, ϕ j t = (c ϕ ij ) tdt. no-trade parallelotop is spanned by a 1,..., a d R d, where a = (a 1,..., a d ) R d d is given by a ij = (γ 1 3εS ) ij = i 3 2p (cs c ϕ c S ) jj (c S ) 1 ij certainty equivalent of utility loss more complicated than in univariate case 23 / 26

The multivariate case The solution ( ) εs S i = (γi x) 3 3γi i x 2 works for γ = (γ 1,..., γ d ) R d d given by 2p 1 γ ij = 3 3εS i (c S c ϕ c S cij S ), ii where d S i, S j t = (c S ij ) tdt and d ϕ i, ϕ j t = (c ϕ ij ) tdt. no-trade parallelotop is spanned by a 1,..., a d R d, where a = (a 1,..., a d ) R d d is given by a ij = (γ 1 3εS ) ij = i 3 2p (cs c ϕ c S ) jj (c S ) 1 ij certainty equivalent of utility loss more complicated than in univariate case 24 / 26

Ingredient of the certainty equivalent of utility loss Covariance matrix of correlated Brownian motion with oblique reflection at the boundary x[, 2] 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 x[, 1] What is the covariance matrix of the stationary law? 25 / 26

26 / 26 Conclusion Shadow price approach is useful to the engineer as well. One can get quite far with regards to explicit formulas for asyptotically optimal portfolios. Rigorous theorems still left to future research.